Question

rewrite the equation in vertex form y = - 2x² + 8x - 6

Explanation:

Step1: Recall vertex - form of quadratic function

The vertex - form of a quadratic function is \(y=a(x - h)^2+k\), where \((h,k)\) is the vertex of the parabola. Given \(y=-2x^{2}+8x - 6\), first factor out the coefficient of \(x^{2}\) from the first two terms. So \(y=-2(x^{2}-4x)-6\).

Step2: Complete the square inside the parentheses

For the quadratic expression \(x^{2}-4x\), we know that \((x - m)^2=x^{2}-2mx + m^{2}\). In \(x^{2}-4x\), if \(-2m=-4\), then \(m = 2\) and \(x^{2}-4x=(x - 2)^{2}-4\). So \(y=-2((x - 2)^{2}-4)-6\).

Step3: Expand and simplify

Expand \(-2((x - 2)^{2}-4)-6\): \(y=-2(x - 2)^{2}+8 - 6\). Then \(y=-2(x - 2)^{2}+2\).

Answer:

\(y=-2(x - 2)^{2}+2\)